SOLUTION: How do I solve this word problem? The polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is 3 times the coefficient o

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: How do I solve this word problem? The polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is 3 times the coefficient o      Log On


   

Question 1045163: How do I solve this word problem?
The polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is 3 times the coefficient of x^2. The remaining coefficient is 2 more than the coefficient of x^3. The sum of the coefficients is -4. Find the polynomial.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Assign variables to coefficients and transcribe literally into equations. Solve the system. Do not let the formal description of the numbers put you offcourse. This is a simple linear system application in three variables.

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The polynomial, ax%5E3%2Bbx%5E2%2Bcx%2Bd

The description of the coefficients, system%28b=a-3%2Cc=3b%2Cd=a%2B2%2Ca%2Bb%2Bc%2Bd=-4%29

system%28a-b=3%2Cc=3b%2Cd=a%2B2%2Ca%2Bb%2Bc%2Bd=-4%29

substituting for c
system%28a-b=3%2Cd=a%2B2%2Ca%2Bb%2B3b%2Bd=-4%29
-
system%28a-b=3%2Cd=a%2B2%2Ca%2B4b%2Bd=-4%29

Substituting for d
system%28a-b=3%2Ca%2B4b%2Ba%2B2=-4%29
system%28a-b=3%2C2a%2B4b=-6%29
system%28a-b=3%2Ca%2B2b=-3%29


Partially ready for Elimination Method...subtract first from second.
2b%2Bb=-3-3 which eliminated "a".
3b=-6
highlight%28b=-2%29


b=a-3
a-3=b
a=b%2B3
a=-2%2B3
highlight%28a=1%29

c=3b
c=3%2A%28-2%29
highlight%28c=-6%29

d=a%2B2
d=1%2B2
highlight%28d=3%29


SUMMARY: The polynomial is highlight_green%28highlight%28x%5E3-2x%5E2-6x%2B3%29%29.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
How do I solve this word problem?
The polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is 3 times the coefficient of x^2. The remaining coefficient is 2 more than the coefficient of x^3. The sum of the coefficients is -4. Find the polynomial.
 You only need ONE (1) variable. Don't confuse yourself with too many unnecessary variables.

Let the coefficient on x%5E3 be a

Then the coefficient on x%5E2 = a - 3

The coefficient on x is: 3(a - 3)

The constant is: a + 2

This gives us: matrix%284%2C1%2C+ax%5E3%2C%28a+-+3%29x%5E2%2C3%28a+-+3%29x%2C%28a+%2B+2%29%29

Since all coefficients sum to  - 4, we get: a + (a - 3) + 3(a - 3) + (a + 2) = - 4

a + a - 3 + 3a - 9 + a + 2 = - 4

6a - 10 = - 4

6a = 6

a = 1

Therefore, matrix%284%2C1%2C+ax%5E3%2C%28a+-+3%29x%5E2%2C3%28a+-+3%29x%2C%28a+%2B+2%29%29 becomes: matrix%284%2C1%2C+1x%5E3%2C%281+-+3%29x%5E2%2C3%281+-+3%29x%2C%281+%2B+2%29%29 ----- Substituting 1 for a

The polynomial is: highlight_green%28x%5E3+-+2x%5E2+-+6x+%2B+3%29

It is that simple...nothing COMPLEX!